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Fluid mechanics is the study of the macroscopic physical behaviour of fluids. Fluids are specifically liquids and gases though some other materials and systems can be described in a similar way. The solution of a fluid dynamic problem typically involves calculating for various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time. Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table:

Continuum mechanics	Solid mechanics: the study of the physics of continuous solids with a defined rest shape.	Elasticity: which describes materials that return to their rest shape after an applied stress.
		Plasticity: which describes materials that permanently deform after a large enough applied stress.	Rheology: the study of materials with both solid and fluid characteristics
	Fluid mechanics	Non-Newtonian fluids
		Newtonian fluids

Fluid mechanics has a wide range of applications. For example, it is used in calculating forces and moments on aircraft, the mass flow of petroleum through pipelines, and in prediction of weather patterns. Fluid mechanics offers a mathematical structure that underlies these practical discipines which often also embrace empirical and semi-empirical laws, derived from flow measurement, to solve practical problems.

Overview of fluid mechanics

Fluid mechanics	Fluid statics
	Fluid dynamics	Laminar flow	Newtonian fluids	Ideal fluids	Incompressible flow
					Compressible flow
				Viscous fluids
					Computational fluid dynamics
					Solutions for specific regimes
			Non-Newtonian fluids	Rheology
		Turbulence

Newtonian versus non-Newtonian fluids

Sir Isaac Newton showed how stress and the rate of change of strain are related in a simple was for many familiar fluids, such as water and air. These Newtonian fluids are characterised by a simple viscosity.

However, some other materials, such as milk and blood, and also some plastic solids, have more complicated non-Newtonian stress-strain behaviours. These are studied in the sub-discipline of rheology.

Fluid phenomena

The following observed fluid phenomena can be characterised and explained using fluid mechanics:

Applications

Fluid dynamics is the study of fluids (liquids and gases) in motion, and the effect of the fluid motion on fluid boundaries, such as solid containers or other fluids. Fluid dynamics is a branch of fluid mechanics, and has a number of subdisciplines, including aerodynamics (the study of gases in motion) and hydrodynamics (liquids in motion). These fields are used in such wide-ranging fields as calculating forces and moments on aircraft, the mass flow of petroleum through pipelines, prediction of weather patterns, and even traffic engineering, where traffic is treated as a continuous flowing fluid.

The continuity assumption

Gases are composed of molecules which collide with one another and solid objects. The continuity assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at infinitely small points, and are assumed to vary continuously from one point to another. The discrete, molecular nature of a fluid is ignored.

Those problems for which the continuity assumption does not give answers of desired accuracy are solved using statistical mechanics. In order to determine whether to use conventional fluid dynamics (a subdiscipline of continuum mechanics) or statistical mechanics, the Knudsen number is evaluated for the problem. Problems with Knudsen numbers at or above unity must be evaluated using statistical mechanics for reliable solutions.

Equations of fluid dynamics

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of momentum (also known as Newton's second law or the balance law), and conservation of energy. These are based on classical mechanics and are modified in relativistic mechanics.

The central equations for fluid dynamics are the Navier-Stokes equations, which are non-linear differential equations that describe the flow of a fluid whose stress depends linearly on velocity and on pressure. The unsimplified equations do not have a general closed-form solution, so they are only of use in computational fluid dynamics. The equations can be simplified in a number of ways. All of the simplifications make the equations easier to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form.

Compressible vs incompressible flow

A fluid problem is called compressible if changes in the density of the fluid have significant effects on the solution. If the density changes have negligible effects on the solution, the fluid is called incompressible and the changes in density are ignored.

In order to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the problem is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. Nearly all problems involving liquids are in this regime and modeled as incompressible.

The incompressible Navier-Stokes equations are simplifications of the Navier-Stokes equations in which the density has been assumed to be constant. These can be used to solve incompressible problems.

Viscous vs inviscid flow

Viscous problems are those in which fluid friction have significant effects on the solution. Problems for which friction can safely be neglected are called inviscid.

The Reynolds number can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. High Reynolds numbers indicate that the inertial forces are more significant than the viscous forces. However, even in high Reynolds number regimes certain problems require that viscosity be included. In particular, problems calculating net forces on bodies (such as wings) should use viscous equations. As illustrated by d'Alembert's paradox, a body in an inviscid fluid will experience no force.

The standard equations of inviscid flow are the Euler equations. Another often used model, especially in computational fluid dynamics, is to use the Euler equations far from the body and the boundary layer equations close to the body.

The Euler equations can be integrated along a streamline to get Bernoulli's equation. When the flow is everywhere irrotational as well as inviscid, Bernoulli's equation can be used to solve the problem.

Steady vs unsteady flow

Another simplification of fluid dynamics equations is to set all changes of fluid properties with time to zero. This is called steady flow, and is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe. Both the Navier-Stokes equations and the Euler equations become simpler when their steady forms are used.

If a problem is incompressible, irrotational, inviscid, and steady, it can be solved using potential flow, governed by Laplace's equation. Problems in this class have elegant solutions which are linear combinations of well-studied elementary flows.

Laminar vs turbulent flow

Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. Mathematically, turbulent flow is often represented via Reynolds decomposition where the flow is broken down into the sum of a steady component and a perturbation component.

It is believed that turbulent flows obey the Navier-Stokes equations. Direct Numerical Simulation (DNS), based on the Navier-Stokes and incompressibility equations, makes it possible to simulate turbulent flows with moderate Reynolds numbers (restrictions depend on the power of computer). The results of DNS agree with the experimental data.

Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Stokes flow is flow at very low Reynold's numbers, such that inertial forces can be neglected compared to viscous forces. The Boussinesq approximation neglects variations in density except to calculate buoyancy forces.